Properties

Label 11.12.a.a
Level $11$
Weight $12$
Character orbit 11.a
Self dual yes
Analytic conductor $8.452$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [11,12,Mod(1,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 11.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.45177498616\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.202533.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 37x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - 4 \beta_{2} - 2 \beta_1 - 131) q^{3} + (17 \beta_{2} + 36 \beta_1 + 952) q^{4} + (90 \beta_{2} + 100 \beta_1 - 2435) q^{5} + ( - 110 \beta_{2} + 251 \beta_1 + 2288) q^{6} + ( - 440 \beta_{2} + 310 \beta_1 - 1694) q^{7} + ( - 404 \beta_1 - 92224) q^{8} + (348 \beta_{2} - 420 \beta_1 - 66066) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - 4 \beta_{2} - 2 \beta_1 - 131) q^{3} + (17 \beta_{2} + 36 \beta_1 + 952) q^{4} + (90 \beta_{2} + 100 \beta_1 - 2435) q^{5} + ( - 110 \beta_{2} + 251 \beta_1 + 2288) q^{6} + ( - 440 \beta_{2} + 310 \beta_1 - 1694) q^{7} + ( - 404 \beta_1 - 92224) q^{8} + (348 \beta_{2} - 420 \beta_1 - 66066) q^{9} + (1540 \beta_{2} - 2245 \beta_1 - 216480) q^{10} + 161051 q^{11} + ( - 35 \beta_{2} - 5908 \beta_1 - 586792) q^{12} + (5808 \beta_{2} - 4478 \beta_1 - 811404) q^{13} + ( - 21110 \beta_{2} - 4186 \beta_1 - 1338320) q^{14} + (19750 \beta_{2} + 6410 \beta_1 - 1920055) q^{15} + ( - 27948 \beta_{2} + 33040 \beta_1 - 737696) q^{16} + ( - 16280 \beta_{2} + 58778 \beta_1 + 4037374) q^{17} + (19668 \beta_{2} + 77010 \beta_1 + 1582944) q^{18} + (142340 \beta_{2} - 66050 \beta_1 - 2863520) q^{19} + ( - 90715 \beta_{2} + 74020 \beta_1 + 13151000) q^{20} + (45716 \beta_{2} - 204662 \beta_1 + 9340474) q^{21} - 161051 \beta_1 q^{22} + ( - 231814 \beta_{2} - 612722 \beta_1 - 1045471) q^{23} + (324456 \beta_{2} + 285852 \beta_1 + 13005696) q^{24} + ( - 1013500 \beta_{2} + \cdots + 19385900) q^{25}+ \cdots + (56045748 \beta_{2} + \cdots - 10639995366) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 393 q^{3} + 2856 q^{4} - 7305 q^{5} + 6864 q^{6} - 5082 q^{7} - 276672 q^{8} - 198198 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 393 q^{3} + 2856 q^{4} - 7305 q^{5} + 6864 q^{6} - 5082 q^{7} - 276672 q^{8} - 198198 q^{9} - 649440 q^{10} + 483153 q^{11} - 1760376 q^{12} - 2434212 q^{13} - 4014960 q^{14} - 5760165 q^{15} - 2213088 q^{16} + 12112122 q^{17} + 4748832 q^{18} - 8590560 q^{19} + 39453000 q^{20} + 28021422 q^{21} - 3136413 q^{23} + 39017088 q^{24} + 58157700 q^{25} + 56471472 q^{26} + 75146805 q^{27} - 10688304 q^{28} - 376441824 q^{29} - 2706000 q^{30} - 313174893 q^{31} + 191457024 q^{32} - 63293043 q^{33} - 574325520 q^{34} - 382305330 q^{35} - 232424784 q^{36} - 454281387 q^{37} + 990724560 q^{38} - 39563700 q^{39} + 411322560 q^{40} - 37614456 q^{41} + 1969231344 q^{42} + 162163386 q^{43} + 459961656 q^{44} + 681234210 q^{45} + 4869127824 q^{46} - 3182498184 q^{47} + 1935867552 q^{48} - 786317721 q^{49} + 2209416000 q^{50} - 899249934 q^{51} - 2346942048 q^{52} - 3000753402 q^{53} - 3684543984 q^{54} - 1176477555 q^{55} - 1153361472 q^{56} - 7959188160 q^{57} + 4842070944 q^{58} - 1843219707 q^{59} + 402214440 q^{60} - 28094112684 q^{61} + 9727390992 q^{62} - 4429082988 q^{63} - 6231589248 q^{64} + 10886398380 q^{65} + 1105454064 q^{66} + 10315312497 q^{67} + 24403080240 q^{68} + 20149986423 q^{69} - 21536889840 q^{70} + 3703071657 q^{71} + 20197140480 q^{72} + 14017034988 q^{73} + 17624567136 q^{74} + 64130925300 q^{75} + 3186205440 q^{76} - 818461182 q^{77} - 32706997968 q^{78} - 8104583058 q^{79} - 11010876000 q^{80} - 41340118797 q^{81} - 93503290704 q^{82} + 26009027946 q^{83} - 20428682736 q^{84} - 13372469550 q^{85} - 16133282880 q^{86} + 51443292384 q^{87} - 44558302272 q^{88} - 17344395051 q^{89} + 65091454560 q^{90} - 65235326952 q^{91} - 220778124696 q^{92} + 66222601959 q^{93} - 243219057312 q^{94} + 170830572000 q^{95} + 75094339584 q^{96} - 7984545237 q^{97} + 176742223680 q^{98} - 31919986098 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 37x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{2} + 6\nu - 52 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -4\nu^{2} + 12\nu + 96 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 8 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + 2\beta _1 + 200 ) / 8 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
6.62795
−5.57381
−0.0541376
−75.6271 −281.520 3671.46 5111.20 21290.6 21831.1 −122777. −97893.2 −386545.
1.2 23.3081 296.237 −1504.73 −13329.8 6904.73 32948.8 −82807.5 −89390.6 −310692.
1.3 52.3190 −407.717 689.274 913.580 −21331.3 −59861.9 −71087.1 −10914.2 47797.6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.12.a.a 3
3.b odd 2 1 99.12.a.a 3
4.b odd 2 1 176.12.a.e 3
5.b even 2 1 275.12.a.a 3
11.b odd 2 1 121.12.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.12.a.a 3 1.a even 1 1 trivial
99.12.a.a 3 3.b odd 2 1
121.12.a.c 3 11.b odd 2 1
176.12.a.e 3 4.b odd 2 1
275.12.a.a 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 4500T_{2} + 92224 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(11))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 4500T + 92224 \) Copy content Toggle raw display
$3$ \( T^{3} + 393 T^{2} + \cdots - 34002261 \) Copy content Toggle raw display
$5$ \( T^{3} + \cdots + 62243294875 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots + 43059126844184 \) Copy content Toggle raw display
$11$ \( (T - 161051)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 14\!\cdots\!59 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 18\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 69\!\cdots\!75 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 98\!\cdots\!23 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 60\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 19\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 45\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 30\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 26\!\cdots\!15 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 79\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 86\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 11\!\cdots\!39 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 20\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 54\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 22\!\cdots\!15 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 28\!\cdots\!93 \) Copy content Toggle raw display
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